On connection between reducibility of an n-ary quasigroup and that of its retracts

Abstract

An $n$-ary operation $Q:S^n\to S$ is called an $n$-ary quasigroup of order $|S|$ if in the equation $x_0=Q(x_1,...,x_n)$ knowledge of any $n$ elements of $x_0,...,x_n$ uniquely specifies the remaining one. An $n$-ary quasigroup $Q$ is (permutably) reducible if $Q(x_1,...,x_n)=P(R(x_{s(1)},...,x_{s(k)}),x_{s(k+1)},...,x_{s(n)})$ where $P$ and $R$ are $(n-k+1)$-ary and $k$-ary quasigroups, $s$ is a permutation, and $1<k<n$. An $m$-ary quasigroup $R$ is called a retract of $Q$ if it can be obtained from $Q$ or one of its inverses by fixing $n-m>0$ arguments. We show that every irreducible $n$-ary quasigroup has an irreducible $(n-1)$-ary or $(n-2)$-ary retract; moreover, if the order is finite and prime, then it has an irreducible $(n-1)$-ary retract. We apply this result to show that all $n$-ary quasigroups of order 5 or 7 whose all binary retracts are isotopic to $Z_5$ or $Z_7$ are reducible for $n>3$. Keywords: $n$-ary quasigroups, retracts, reducibility, latin hypercubes

English: 19pp; Russian: 20pp. V.2: case n=4 added, Russian translation added, title changed (old title: On reducibility of n-ary quasigroups, II)

References in corpus (5)

• n-Ary quasigroups of order 4
• Asymptotics for the number of n-quasigroups of order 4
• On reconstructing reducible n-ary quasigroups and switching subquasigroups
• On irreducible n-ary quasigroups with reducible retracts
• On reducibility of n-ary quasigroups

Cited by in corpus (4)

• Constructions of transitive latin hypercubes
• Every latin hypercube of order 5 has transversals
• On a connection between the switching separability of a graph and that of its subgraphs
• On a test on switching separability of graphs modulo $q$